Monday, September 07, 2009

Analyzing NLHE:TAP Concept 11

I'm commenting on the sixty concepts at the end of No Limit Hold 'em: Theory and Practice by David Sklansky and Ed Miller. Let me apologize in advance: this one is a little more mathematical and a little less conclusive than most.

Concept No. 11: A big bet is the most relevant and accurate information at the table.

Once again, I think Sklansky and Miller's advice here is probably bad, but this one is particularly difficult to analyze because it's unclear what is meant by "relevant and accurate information." In any case, this concept's claim is far too broadly stated, and only holds in some circumstances.

If we take the claim in the most literal sense, this concept is not always true. Infinite counterexamples exist. Perhaps you know that your opponent always makes big bets; in such a case, the big bet provides essentially no information at all. Or, sometimes you may have the nuts, which is probably the most "relevant" factor if you're trying to figure out how to proceed with the hand. As far as "accurate" information goes, usually preflop play is most accurate, since you will have observed your opponent in many more preflop situations than postflop situations. Of course, all of this varies from player to player. It seems overly bold for S+M to make a claim that a big bet is (presumably, always) the most relevant and accurate.

If we're not to take this concept literally, let's consider whether it is at least good advice in a figurative sense. This concept seems suggest that players should respect the big bet more than they do, and thus fold to it more. Do players tend to underestimate the importance of a big bet when trying to determine their opponent's possible hand?
Human nature actually suggests the exact opposite; there is a phenomenon called the "base rate fallacy" that seems like it could apply here. It has been established that people tend to discount base rates (which could correspond to your read of your opponent before the big bet), and focus only on the most immediate and salient information (such as the big bet) when estimating likelihoods. Are poker players similarly likely to focus to much on the most immediate information? I think sometimes yes, sometimes no. There are certainly a lot of players who are not sensitive enough to bet sizes, and for these players, taking this concept to heart is probably a good idea. I'm not sure that most players fall into this category, though.

In their explanation of this concept, Sklansky and Miller say "all information from the past takes a back seat to the fact that they've made a big bet." I think a fair way to interpret this is "the probability your opponent holds the nuts, or nearly the nuts, given that he just made a big bet, is greater than the likelihood that he did NOT hold the nuts before he made that big bet." Mathematically, we can write this as:
P(nuts|big bet) > p(nuts' before the big bet). That is: P(nuts|big bet) > p(nuts').
(Note that nuts' means "not the nuts.")
I'm fairly confident that this is not always true, although it is might be true if your opponent is a very good player. I think that Sklansky and Miller might have succumbed to the base rate fallacy and are underestimating the importance of their initial read of their opponent's preflop tendencies. Let's do some Bayesian analysis of
the example they give in their explanation of this concept.

Suppose we are facing a big bet and we think our opponent is either bluffing or holding the nuts. Before he made the big bet, suppose we thought that the way he played the hand meant he had only a 0.05% chance of holding the nuts. This seems reasonable in S+M's example where your opponent is very tight, raised pre-flop, and the nuts is 74
(actually, .05% seems a bit too high, but let's give S+M the benefit of the doubt). Now let's say we think our opponent would make a big bet 80% of the time if he held 74. Otherwise, he would make the same big bet 2% of the time as a bluff. The base rate fallacy would be to think that your opponent would now be holding the nuts 80/82 times, or about 97.6% of the time. Of course, this is way too high because it ignores base rates. In this example, we can use Bayes' theorem to find the actual probability that you are up against the nuts. Bayes gives us the following equation:
P(nuts|big bet) = P(big bet|nuts)*P(nuts) / (P(big bet|nuts)*P(nuts)
P(big bet|nuts')*P(nuts'))
From our assumptions above, we have:

P(big bet|nuts) = .8
P(nuts) = .0005
big bet|nuts') = .02
(nuts') = .9999

and so:

P(big bet|nuts)*P(nuts) = .0004
(big bet|nuts')*P(nuts') = .019998

P(nuts|big bet) = .0004 / (.0004 + .019998) = .0004/.020398
= .0196
So, under these assumptions, our opponent holds the nuts less than 2% of the time when he makes a big bet! You may quibble with the rates I chose, but even if our calculations are off by a factor of 10, our opponent will still only be holding a winning hand 20% of the time, and bluffing 80%, making it correct to call any sized bet.

If you don't like the rates I chose, I guess it's most likely because you think that even very tight players tend to hold 74 on the river more than .05% of the time in this situation (the situation: he raised preflop, and 74 is the nuts on the river). I admit, this is a tricky one to estimate, but we can do another Bayesian inference to examine my .05% claim.
That is, we want to know if .05% is a reasonable approximation for how often a tight, pre-flop raiser holds 74 on the river in this situation. That is, we want to find P(74|saw river).

First of all, let's assume our "tight" player raises preflop with 74 only 1% of the time when dealt to him. Also, assuming we don't have a 7 or a 4 in our hand, our opponent will have been dealt 74 only 1.3% of the time. So, P(74) = .013*.01 = .00013.

Second of all, we need to approximate how often 74 will see the river in this situation. Remember, the situation is: 74 has already raised preflop and seen a flop that gives him a draw to the nuts. IN order to be conservative and to simplify the math, let's assume he will never fold before the river in this situation. So, P(saw river|74) = 1.

Third of all, we need to approximate how often a player would have seen the river in this situation given that he raised preflop holding anything. This is another tough one. I think this is likely over 50%, but let's be conservative and choose 26%, which will also help out with the arithmetic.
So, assume P(saw river) = .26. Bayes theorem then gives us:
P(74|saw river) = P(saw river|74)*P(74) / P(saw river)
= 1*.00013/.26 = .0005 = .05%
Well, this is precisely the number I used above, and I was using conservative numbers in this example. Of course, you could still contend that my numbers or other assumptions are unreasonable, but I have at least convinced myself that in Sklansky and Miller's example, our opponent's big bet usually will not indicate that he has the nuts.


Dave said...

Hey Keith, been following (and mostly agreeing with) your NLHE TAP analysis. I "loaned" out that book some time back so I can't reference it, but as I recall those concepts were the most relevant and helpful parts of that book.

I find this concept to be one of the most true. I recall only that they say the big bet is RELEVANT INFO, whereas you seem to be taking them as saying it means the nuts are out there.

If we soften their stance to say it's just info, then the info might be that our opponent loves his hand, and if we're holding the nuts, we no longer have to wonder how much we can charge him.

But more often than not it seems mean that they have it, whatever "it" is on that particular board. Playing $5 blinds and you reraised somebody to $50 preflop on with KK and then see a 622 board? If the betting gets nutty, yeah, start putting them on 66 or A2s or something along those lines.

Doesn't necessarily mean one has to fold, but you might want to start figuring out how to keep some of your chips out the middle.

I'm not great at math, nor sophisticated enough to illustrate the point, but in my experience even when the odds are very, very tiny that our opponent made it this far with 74, or 96, or whatever crushes our holding, the odds go WAY up when the betting gets deep. So if normal odds that they have 74 are like 5%, it's now like 80%.

I'll try to express it in a real world example. Let's say the percentage of Americans that believe in God is 70%. But now you take that same poll inside of a Catholic church. You're going to have a lot more than 7 out of 10 believers. There is a term for this in statistics that I'm not smart enough to know, but basically you have a warped distribution sample.

Yikes, this got wordy fast...

Keith said...
This comment has been removed by the author.
Keith said...

Dave, you make some good points. The big bet certainly is relevant information, I'm just not so sure if it is the MOST relevant or the MOST accurate, as this concept says.

You also point out that we often should be worried about strong hands that are not necessarily the nuts. This is a good point. I probably got a little too carried away analyzing the example in the book without addressing the actual concept directly (which is why I apologized at the beginning of my post).

In the example in the book, S+M describe a player who "raised preflop, and he's as tight a player as they come." Then they say that the big bet means he very possibly could have 74 if it makes a big hand. I tried using Bayes' theorem to show that it is still very unlikely this guy has 74, and I think I succeeded (to my own satisfaction, at least). That doesn't necessarily invalidate the concept, but it may invalidate S+M's reasoning. I think the burden of proof should be on Sklansky and Miller here when they make such grand proclamations as "A big bet is the most relevant and accurate information available." It may be true, but their explanation is, I think, wrong.

By the way, I think you were referring to a "selection bias," but I don't think my analysis had this particular problem.

Anonymous said...

Once again I agree with you, Keith. This advice might sometimes be true, but that's about it. In fact I have noticed that some players like to bet big when they are bluffing and small when they are value betting. A lot of people like to make big bets with a hand like J9 on a 9-high board. I guess that doesn't directly contradict S&M, since they only use the word "information", but it seems like they are implying that a big bet should warn you strongly that someone might have a huge hand.

I must admit, though, that I can think of many times when I have called a big bet thinking "He couldn't have a huge hand, otherwise why would he bet so much?" only to find out that the guy did have a great hand. But I doubt any general advice will help here. It's just like everything in poker. You have to make educated guesses and try to evaluate your opponent. That's the fun part. It can be a little annoying that books like S&M's try to oversimplify things so much.

Anonymous said...

By the way, I was just looking at that link you have called "law school dropout". It's so interesting! He is so likeable, and he in many ways seems so honest, thoughtful, and insightful, but then in the end it is staggering what a complete and utter hypocrite he is. Also, although he sometimes seems very honest, I wonder about certain things. He seems a bit reluctant, for instance, to admit that he might not be the greatest thing ever to hit the poker world, and he analyzes his win streaks and his losing streaks separately, and interprets the losing streaks as abberrations, posts less often about poker when he is losing, etc. For those of us who have jobs but have sometimes wondered what it would be like to quit everything and just play poker, his blog leaves one very confused.

Anonymous said...

Those last 2 posts were from me, Rick. I still can't remember my google password and am too lazy to figure out how to get a new one.

Anonymous said...

I recently came accross your blog and have been reading along. I thought I would leave my first comment. I dont know what to say except that I have enjoyed reading. Nice blog. I will keep visiting this blog very often.